Hopf-Hopf分叉的随机动力学研究

分叉理论一直是非线性动力系统的核心问题，其在自治系统与周期系统已有了较完善的理论体系，而在随机系统的理论研究相对较少。由于随机现象的普遍性，往往需要建立更加符合实际问题的随机模型，来揭示随机扰动下系统稳定性的演变规律。目前随机系统的分叉理论多集中于余维一分叉，尚未考虑到高余维分叉参数受到随机扰动的情形。本项目拟突破随机扰动下余维二 Hopf-Hopf（HH）分叉的基础理论研究。从该分叉的四维规范形系统入手，运用分叉理论、平均场理论等微分方程经典方法，分析其随机动力学行为。然后通过数值模拟直观展示随机扰动后系统可能出现的新的分叉现象，探索系统有极限环、异宿环以及环爆炸现象时随机参数的可控范围。此外，本项目将把所得理论结果应用于具体的模型，明确随机扰动在模型发展过程中产生的反馈机制，为物理电路的调控、金融市场的价格预测等提供新的视角。

Bifurcation theory has always been the core issue of nonlinear dynamic systems, and it has been well-developed in autonomous and periodic systems. Bifurcations in stochastic systems were relatively less studied. Due to the universality of stochastic phenomena, it is necessary to establish a more practical stochastic model to reveal the evolution law of the stability of the model under stochastic perturbation. At present, the existing theories mostly focus on the codimension one bifurcation under stochastic perturbation, and have not yet taken into account the case of high codimension bifurcation. This project intends to break through the basic theoretical research of codimension two Hopf-Hopf (HH) bifurcation under stochastic perturbation. Starting from a four-dimensional normal form system of HH bifurcation, the classical methods of differential equations, including bifurcation theory, mean field theory and so on, are used to analyze its stochastic dynamic behaviors. Then through numerical simulation, the new bifurcation phenomena that may occur in the system under stochastic perturbation are visually displayed, and the controllable ranges of stochastic parameters when the system has limit cycles, heteroclinic orbit and cycle blow-up are explored. In addition, the project will apply the obtained theoretical results to specific models, clarify the feedback mechanism generated by stochastic perturbation in the development process of the model, and provide a new insight for the regulation of physical circuits, the price prediction of financial markets and so on.